Iterative reconstruction

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Example showing differences between filtered backprojection (right half) and iterative reconstruction method (left half) FBP Iter single.jpg
Example showing differences between filtered backprojection (right half) and iterative reconstruction method (left half)

Iterative reconstruction refers to iterative algorithms used to reconstruct 2D and 3D images in certain imaging techniques. For example, in computed tomography an image must be reconstructed from projections of an object. Here, iterative reconstruction techniques are usually a better, but computationally more expensive alternative to the common filtered back projection (FBP) method, which directly calculates the image in a single reconstruction step. [1] In recent research works, scientists have shown that extremely fast computations and massive parallelism is possible for iterative reconstruction, which makes iterative reconstruction practical for commercialization. [2]

Contents

Basic concepts

CT scan using iterative reconstruction (left) versus filtered backprojection (right) CT scan Iterative reconstruction (left) versus filtered backprojection (right).jpg
CT scan using iterative reconstruction (left) versus filtered backprojection (right)

The reconstruction of an image from the acquired data is an inverse problem. Often, it is not possible to exactly solve the inverse problem directly. In this case, a direct algorithm has to approximate the solution, which might cause visible reconstruction artifacts in the image. Iterative algorithms approach the correct solution using multiple iteration steps, which allows to obtain a better reconstruction at the cost of a higher computation time.

There are a large variety of algorithms, but each starts with an assumed image, computes projections from the image, compares the original projection data and updates the image based upon the difference between the calculated and the actual projections.

Algebraic reconstruction

The Algebraic Reconstruction Technique (ART) was the first iterative reconstruction technique used for computed tomography by Hounsfield.

iterative Sparse Asymptotic Minimum Variance

The iterative Sparse Asymptotic Minimum Variance algorithm is an iterative, parameter-free superresolution tomographic reconstruction method inspired by compressed sensing, with applications in synthetic-aperture radar, computed tomography scan, and magnetic resonance imaging (MRI).

Statistical reconstruction

There are typically five components to statistical iterative image reconstruction algorithms, e.g. [3]

  1. An object model that expresses the unknown continuous-space function that is to be reconstructed in terms of a finite series with unknown coefficients that must be estimated from the data.
  2. A system model that relates the unknown object to the "ideal" measurements that would be recorded in the absence of measurement noise. Often this is a linear model of the form , where represents the noise.
  3. A statistical model that describes how the noisy measurements vary around their ideal values. Often Gaussian noise or Poisson statistics are assumed. Because Poisson statistics are closer to reality, it is more widely used.
  4. A cost function that is to be minimized to estimate the image coefficient vector. Often this cost function includes some form of regularization. Sometimes the regularization is based on Markov random fields.
  5. An algorithm, usually iterative, for minimizing the cost function, including some initial estimate of the image and some stopping criterion for terminating the iterations.

Learned Iterative Reconstruction

In learned iterative reconstruction, the updating algorithm is learned from training data using techniques from machine learning such as convolutional neural networks, while still incorporating the image formation model. This typically gives faster and higher quality reconstructions and has been applied to CT [4] and MRI reconstruction. [5]

Advantages

A single frame from a real-time MRI (rt-MRI) movie of a human heart. a) direct reconstruction b) iterative (nonlinear inverse) reconstruction Heart-direct-vs-iterative-reconstruction.png
A single frame from a real-time MRI (rt-MRI) movie of a human heart. a) direct reconstruction b) iterative (nonlinear inverse) reconstruction

The advantages of the iterative approach include improved insensitivity to noise and capability of reconstructing an optimal image in the case of incomplete data. The method has been applied in emission tomography modalities like SPECT and PET, where there is significant attenuation along ray paths and noise statistics are relatively poor.

Statistical, likelihood-based approaches: Statistical, likelihood-based iterative expectation-maximization algorithms [7] [8] are now the preferred method of reconstruction. Such algorithms compute estimates of the likely distribution of annihilation events that led to the measured data, based on statistical principle, often providing better noise profiles and resistance to the streak artifacts common with FBP. Since the density of radioactive tracer is a function in a function space, therefore of extremely high-dimensions, methods which regularize the maximum-likelihood solution turning it towards penalized or maximum a-posteriori methods can have significant advantages for low counts. Examples such as Ulf Grenander's Sieve estimator [9] [10] or Bayes penalty methods, [11] [12] or via I.J. Good's roughness method [13] [14] may yield superior performance to expectation-maximization-based methods which involve a Poisson likelihood function only.

As another example, it is considered superior when one does not have a large set of projections available, when the projections are not distributed uniformly in angle, or when the projections are sparse or missing at certain orientations. These scenarios may occur in intraoperative CT, in cardiac CT, or when metal artifacts [15] [16] require the exclusion of some portions of the projection data.

In Magnetic Resonance Imaging it can be used to reconstruct images from data acquired with multiple receive coils and with sampling patterns different from the conventional Cartesian grid [17] and allows the use of improved regularization techniques (e.g. total variation) [18] or an extended modeling of physical processes [19] to improve the reconstruction. For example, with iterative algorithms it is possible to reconstruct images from data acquired in a very short time as required for real-time MRI (rt-MRI). [6]

In Cryo Electron Tomography, where the limited number of projections are acquired due to the hardware limitations and to avoid the biological specimen damage, it can be used along with compressive sensing techniques or regularization functions (e.g. Huber function) to improve the reconstruction for better interpretation. [20]

Here is an example that illustrates the benefits of iterative image reconstruction for cardiac MRI. [21]

See also

Related Research Articles

<span class="mw-page-title-main">Positron emission tomography</span> Medical imaging technique

Positron emission tomography (PET) is a functional imaging technique that uses radioactive substances known as radiotracers to visualize and measure changes in metabolic processes, and in other physiological activities including blood flow, regional chemical composition, and absorption. Different tracers are used for various imaging purposes, depending on the target process within the body.

<span class="mw-page-title-main">Single-photon emission computed tomography</span> Nuclear medicine tomographic imaging technique

Single-photon emission computed tomography is a nuclear medicine tomographic imaging technique using gamma rays. It is very similar to conventional nuclear medicine planar imaging using a gamma camera, but is able to provide true 3D information. This information is typically presented as cross-sectional slices through the patient, but can be freely reformatted or manipulated as required.

<span class="mw-page-title-main">Tomography</span> Imaging by sections or sectioning using a penetrative wave

Tomography is imaging by sections or sectioning that uses any kind of penetrating wave. The method is used in radiology, archaeology, biology, atmospheric science, geophysics, oceanography, plasma physics, materials science, cosmochemistry, astrophysics, quantum information, and other areas of science. The word tomography is derived from Ancient Greek τόμος tomos, "slice, section" and γράφω graphō, "to write" or, in this context as well, "to describe." A device used in tomography is called a tomograph, while the image produced is a tomogram.

<span class="mw-page-title-main">Electrical impedance tomography</span> Noninvasive type of medical imaging

Electrical impedance tomography (EIT) is a noninvasive type of medical imaging in which the electrical conductivity, permittivity, and impedance of a part of the body is inferred from surface electrode measurements and used to form a tomographic image of that part. Electrical conductivity varies considerably among various biological tissues or the movement of fluids and gases within tissues. The majority of EIT systems apply small alternating currents at a single frequency, however, some EIT systems use multiple frequencies to better differentiate between normal and suspected abnormal tissue within the same organ.

<span class="mw-page-title-main">Tomographic reconstruction</span> Estimate object properties from a finite number of projections

Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security.

Super-resolution imaging (SR) is a class of techniques that enhance (increase) the resolution of an imaging system. In optical SR the diffraction limit of systems is transcended, while in geometrical SR the resolution of digital imaging sensors is enhanced.

<span class="mw-page-title-main">Neuroimaging</span> Set of techniques to measure and visualize aspects of the nervous system

Neuroimaging is the use of quantitative (computational) techniques to study the structure and function of the central nervous system, developed as an objective way of scientifically studying the healthy human brain in a non-invasive manner. Increasingly it is also being used for quantitative research studies of brain disease and psychiatric illness. Neuroimaging is highly multidisciplinary involving neuroscience, computer science, psychology and statistics, and is not a medical specialty. Neuroimaging is sometimes confused with neuroradiology.

<span class="mw-page-title-main">Maximum intensity projection</span>

In scientific visualization, a maximum intensity projection (MIP) is a method for 3D data that projects in the visualization plane the voxels with maximum intensity that fall in the way of parallel rays traced from the viewpoint to the plane of projection. This implies that two MIP renderings from opposite viewpoints are symmetrical images if they are rendered using orthographic projection.

Compressed sensing is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Nyquist–Shannon sampling theorem. There are two conditions under which recovery is possible. The first one is sparsity, which requires the signal to be sparse in some domain. The second one is incoherence, which is applied through the isometric property, which is sufficient for sparse signals.

In statistics, sieve estimators are a class of non-parametric estimators which use progressively more complex models to estimate an unknown high-dimensional function as more data becomes available, with the aim of asymptotically reducing error towards zero as the amount of data increases. This method is generally attributed to Ulf Grenander.

<span class="mw-page-title-main">Optical projection tomography</span>

Optical projection tomography is a form of tomography involving optical microscopy. The OPT technique is sometimes referred to as Optical Computed Tomography (optical-CT) and Optical Emission Computed Tomography (optical-ECT) in the literature, to address the fact that the technique bears similarity to X-ray computed tomography (CT) and single-photon emission computed tomography (SPECT).

Medical image computing (MIC) is an interdisciplinary field at the intersection of computer science, information engineering, electrical engineering, physics, mathematics and medicine. This field develops computational and mathematical methods for solving problems pertaining to medical images and their use for biomedical research and clinical care.

<span class="mw-page-title-main">Frank Natterer</span> German mathematician

Frank Natterer is a German mathematician. He was born in Wangen im Allgäu, Germany. Natterer pioneered and shaped the field of mathematical methods in imaging including computed tomography (CT), magnetic resonance imaging (MRI) and ultrasonic imaging.

Simultaneous algebraic reconstruction technique (SART) is a computerized tomography (CT) imaging algorithm useful in cases when the projection data is limited; it was proposed by Anders Andersen and Avinash Kak in 1984. It generates a good reconstruction in just one iteration and it is superior to standard algebraic reconstruction technique (ART).

<span class="mw-page-title-main">Michael I. Miller</span> American biomedical engineer and neuroscientist

Michael Ira Miller is an American-born biomedical engineer and data scientist, and the Bessie Darling Massey Professor and Director of the Johns Hopkins University Department of Biomedical Engineering. He worked with Ulf Grenander in the field of Computational Anatomy as it pertains to neuroscience, specializing in mapping the brain under various states of health and disease by applying data derived from medical imaging. Miller is the director of the Johns Hopkins Center for Imaging Science, Whiting School of Engineering and codirector of Johns Hopkins Kavli Neuroscience Discovery Institute. Miller is also a Johns Hopkins University Gilman Scholar.

Electrical capacitance volume tomography (ECVT) is a non-invasive 3D imaging technology applied primarily to multiphase flows. It was first introduced by W. Warsito, Q. Marashdeh, and L.-S. Fan as an extension of the conventional electrical capacitance tomography (ECT). In conventional ECT, sensor plates are distributed around a surface of interest. Measured capacitance between plate combinations is used to reconstruct 2D images (tomograms) of material distribution. In ECT, the fringing field from the edges of the plates is viewed as a source of distortion to the final reconstructed image and is thus mitigated by guard electrodes. ECVT exploits this fringing field and expands it through 3D sensor designs that deliberately establish an electric field variation in all three dimensions. The image reconstruction algorithms are similar in nature to ECT; nevertheless, the reconstruction problem in ECVT is more complicated. The sensitivity matrix of an ECVT sensor is more ill-conditioned and the overall reconstruction problem is more ill-posed compared to ECT. The ECVT approach to sensor design allows direct 3D imaging of the outrounded geometry. This is different than 3D-ECT that relies on stacking images from individual ECT sensors. 3D-ECT can also be accomplished by stacking frames from a sequence of time intervals of ECT measurements. Because the ECT sensor plates are required to have lengths on the order of the domain cross-section, 3D-ECT does not provide the required resolution in the axial dimension. ECVT solves this problem by going directly to the image reconstruction and avoiding the stacking approach. This is accomplished by using a sensor that is inherently three-dimensional.

Computational imaging is the process of indirectly forming images from measurements using algorithms that rely on a significant amount of computing. In contrast to traditional imaging, computational imaging systems involve a tight integration of the sensing system and the computation in order to form the images of interest. The ubiquitous availability of fast computing platforms, the advances in algorithms and modern sensing hardware is resulting in imaging systems with significantly enhanced capabilities. Computational Imaging systems cover a broad range of applications include computational microscopy, tomographic imaging, MRI, ultrasound imaging, computational photography, Synthetic Aperture Radar (SAR), seismic imaging etc. The integration of the sensing and the computation in computational imaging systems allows for accessing information which was otherwise not possible. For example:

Superiorization is an iterative method for constrained optimization. It is used for improving the efficacy of an iterative method whose convergence is resilient to certain kinds of perturbations. Such perturbations are designed to "force" the perturbed algorithm to produce more useful results for the intended application than the ones that are produced by the original iterative algorithm. The perturbed algorithm is called the superiorized version of the original unperturbed algorithm. If the original algorithm is computationally efficient and useful in terms of the target application and if the perturbations are inexpensive to calculate, the method may be used to steer iterates without additional computation cost.

Deep learning in photoacoustic imaging

Deep learning in photoacoustic imaging combines the hybrid imaging modality of photoacoustic imaging (PA) with the rapidly evolving field of deep learning. Photoacoustic imaging is based on the photoacoustic effect, in which optical absorption causes a rise in temperature, which causes a subsequent rise in pressure via thermo-elastic expansion. This pressure rise propagates through the tissue and is sensed via ultrasonic transducers. Due to the proportionality between the optical absorption, the rise in temperature, and the rise in pressure, the ultrasound pressure wave signal can be used to quantify the original optical energy deposition within the tissue.

References

  1. ↑ Herman, G. T., Fundamentals of computerized tomography: Image reconstruction from projection, 2nd edition, Springer, 2009
  2. ↑ Wang, Xiao; Sabne, Amit; Kisner, Sherman; Raghunathan, Anand; Bouman, Charles; Midkiff, Samuel (2016-01-01). "High performance model based image reconstruction". Proceedings of the 21st ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming. PPoPP '16. New York, NY, USA: ACM. pp. 2:1–2:12. doi:10.1145/2851141.2851163. ISBN   9781450340922. S2CID   16569156.
  3. ↑ Fessler J A (1994). "Penalized weighted least-squares image reconstruction for positron emission tomography" (PDF). IEEE Transactions on Medical Imaging. 13 (2): 290–300. doi:10.1109/42.293921. hdl: 2027.42/85851 . PMID   18218505.
  4. ↑ Adler, J.; Öktem, O. (2018). "Learned Primal-dual Reconstruction". IEEE Transactions on Medical Imaging. PP (99): 1322–1332. arXiv: 1707.06474 . doi:10.1109/tmi.2018.2799231. ISSN   0278-0062. PMID   29870362. S2CID   26897002.
  5. ↑ Hammernik, Kerstin; Klatzer, Teresa; Kobler, Erich; Recht, Michael P.; Sodickson, Daniel K.; Pock, Thomas; Knoll, Florian (2018). "Learning a variational network for reconstruction of accelerated MRI data". Magnetic Resonance in Medicine. 79 (6): 3055–3071. arXiv: 1704.00447 . doi:10.1002/mrm.26977. ISSN   1522-2594. PMC   5902683 . PMID   29115689.
  6. 1 2 Uecker M, Zhang S, Voit D, Karaus A, Merboldt KD, Frahm J (2010a). "Real-time MRI at a resolution of 20 ms" (PDF). NMR Biomed. 23 (8): 986–994. doi:10.1002/nbm.1585. hdl: 11858/00-001M-0000-0012-D4F9-7 . PMID   20799371. S2CID   8268489.
  7. ↑ Carson, Lange; Richard Carson (1984). "EM reconstruction algorithm for emission and transmission tomography". Journal of Computer Assisted Tomography. 8 (2): 306–316. PMID   6608535.
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  9. ↑ Snyder, Donald L.; Miller, Michael I. (1985). "On the Use of the Method of Sieves for Positron Emission Tomography". IEEE Transactions on Medical Imaging. NS-32(5) (5): 3864–3872. Bibcode:1985ITNS...32.3864S. doi:10.1109/TNS.1985.4334521. S2CID   2112617.
  10. ↑ Snyder, D.L.; Miller, M.I.; Thomas, L.J.; Politte, D.G. (1987). "Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography". IEEE Transactions on Medical Imaging. 6 (3): 228–238. doi:10.1109/tmi.1987.4307831. PMID   18244025. S2CID   30033603.
  11. ↑ Geman, Stuart; McClure, Donald E. (1985). "Bayesian image analysis: An application to single photon emission tomography" (PDF). Proceedings Amererican Statistical Computing: 12–18.
  12. ↑ Green, Peter J. (1990). "Bayesian Reconstructions for Emission Tomography Data Using a Modified EM Algorithm". IEEE Transactions on Medical Imaging. 9 (1): 84–93. CiteSeerX   10.1.1.144.8671 . doi:10.1109/42.52985. PMID   18222753.
  13. ↑ Miller, Michael I.; Snyder, Donald L. (1987). "The role of likelihood and entropy in incomplete data problems: Applications to estimating point-process intensites and toeplitz constrained covariance estimates". Proceedings of the IEEE. 5 (7): 3223–3227. doi:10.1109/PROC.1987.13825. S2CID   23733140.
  14. ↑ Miller, Michael I.; Roysam, Badrinath (April 1991). "Bayesian image reconstruction for emission tomography incorporating Good's roughness prior on massively parallel processors". Proceedings of the National Academy of Sciences of the United States of America. 88 (8): 3223–3227. Bibcode:1991PNAS...88.3223M. doi: 10.1073/pnas.88.8.3223 . PMC   51418 . PMID   2014243.
  15. ↑ Wang, G.E.; Snyder, D.L.; O'Sullivan, J.A.; Vannier, M.W. (1996). "Iterative deblurring for CT metal artifact reduction". IEEE Transactions on Medical Imaging. 15 (5): 657–664. doi:10.1109/42.538943. PMID   18215947.
  16. ↑ Boas FE, Fleischmann D (2011). "Evaluation of two iterative techniques for reducing metal artifacts in computed tomography". Radiology. 259 (3): 894–902. doi:10.1148/radiol.11101782. PMID   21357521. Archived from the original on 2011-12-01.
  17. ↑ Pruessmann K. P., Weiger M., Börnert P., Boesiger P. (2001). "Advances in sensitivity encoding with arbitrary k-space trajectories". Magnetic Resonance in Medicine. 46 (4): 638–651. doi: 10.1002/mrm.1241 . PMID   11590639.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  18. ↑ Block K. T., Uecker M., Frahm J. (2007). "Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint". Magnetic Resonance in Medicine. 57 (6): 1086–1098. doi:10.1002/mrm.21236. hdl: 11858/00-001M-0000-0012-E2A3-7 . PMID   17534903. S2CID   16396739.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  19. ↑ Fessler J (2010). "Model-based Image Reconstruction for MRI". IEEE Signal Processing Magazine. 27 (4): 81–89. Bibcode:2010ISPM...27...81F. doi:10.1109/msp.2010.936726. PMC   2996730 . PMID   21135916.
  20. ↑ Albarqouni, Shadi; Lasser, Tobias; Alkhaldi, Weaam; Al-Amoudi, Ashraf; Navab, Nassir (2015-01-01). "Gradient Projection for Regularized Cryo-Electron Tomographic Reconstruction". In Gao, Fei; Shi, Kuangyu; Li, Shuo (eds.). Computational Methods for Molecular Imaging. Lecture Notes in Computational Vision and Biomechanics. Vol. 22. Springer International Publishing. pp. 43–51. doi:10.1007/978-3-319-18431-9_5. ISBN   978-3-319-18430-2.
  21. ↑ I Uyanik, P Lindner, D Shah, N Tsekos I Pavlidis (2013) Applying a Level Set Method for Resolving Physiologic Motions in Free-Breathing and Non-gated Cardiac MRI. FIMH, 2013, "Computational Physiology Lab" (PDF). Archived from the original (PDF) on 2018-07-22. Retrieved 2013-10-01.

[1] [2]

  1. ↑ Bruyant P.P. (2002). "Analytic and iterative reconstruction algorithms in SPECT". Journal of Nuclear Medicine. 43 (10): 1343–1358. PMID   12368373.
  2. ↑ Grishentcev A. Jr (2012). "Effective compression of images on the basis of differential analysis" (PDF). Journal of Radio Electronics. 11: 1–42.
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